Geordie Richards

On Unique Ergodicity in Nonlinear Stochastic Partial Differential Equations

We illustrate how the notion of asymptotic coupling provides a flexible and intuitive framework for proving the uniqueness of invariant measures for a variety of stochastic partial differential equations whose deterministic counterpart possesses a finite number of determining modes. Examples exhibiting parabolic and hyperbolic structure are studied in detail. In the later situation we also present a simple framework for establishing the existence of invariant measures when the usual approach relying on the Krylov–Bogolyubov procedure and compactness fails.

Asymptotic Analysis for Randomly Forced MHD

We consider the three-dimensional magnetohydrodynamics (MHD) equations in the presence of a spatially degenerate stochastic forcing as a model for magnetostrophic turbulence in the Earths fluid core. We examine the multi-parameter singular limit of vanishing Rossby number

Ergodic and mixing properties of the Boussinesq equations with a degenerate random forcing

\epsilon

On invariant Gibbs measures for the generalized KdV equations

and magnetic Reynolds number

Ergodicity in randomly forced Rayleigh–Bénard convection

\delta

Large Prandtl Number Asymptotics in Randomly Forced Turbulent Convection

, and establish that: (i) the limiting stochastically driven active scalar equation (with

Error propagation dynamics of PIV-based pressure field calculation (3): What is the minimum resolvable pressure in a reconstructed field?

\epsilon =\delta=0

Assessing the Limitations of Effective Number of Samples for Finding the Uncertainty of the Mean of Correlated Data

) possesses a unique ergodic invariant measure, and (ii) any suitable sequence of statistically invariant states of the full MHD system converge weakly, as

Hydrodynamic stability in the presence of a stochastic forcing:a case study in convection

\epsilon,\delta \rightarrow 0

Error Propagation dynamics: from PIV-based pressure reconstruction to vorticity field calculation

, to the unique invariant measure of the limit equation. This latter convergence result does not require any conditions on the relative rates at which